erdsze2lem2 - Mathbox for Mario Carneiro

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Theorem erdsze2lem2 32453

Description: Lemma for erdsze2 32454. (Contributed by Mario Carneiro, 22-Jan-2015.)

**Hypotheses**
Ref	Expression
erdsze2.r	⊢ (𝜑 → 𝑅 ∈ ℕ)
erdsze2.s	⊢ (𝜑 → 𝑆 ∈ ℕ)
erdsze2.f	⊢ (𝜑 → 𝐹:𝐴–1-1→ℝ)
erdsze2.a	⊢ (𝜑 → 𝐴 ⊆ ℝ)
erdsze2lem.n	⊢ 𝑁 = ((𝑅 − 1) · (𝑆 − 1))
erdsze2lem.l	⊢ (𝜑 → 𝑁 < (♯‘𝐴))
erdsze2lem.g	⊢ (𝜑 → 𝐺:(1...(𝑁 + 1))–1-1→𝐴)
erdsze2lem.i	⊢ (𝜑 → 𝐺 Isom < , < ((1...(𝑁 + 1)), ran 𝐺))

**Assertion**
Ref	Expression
erdsze2lem2	⊢ (𝜑 → ∃𝑠 ∈ 𝒫 𝐴((𝑅 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , < (𝑠, (𝐹 “ 𝑠))) ∨ (𝑆 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , ^◡ < (𝑠, (𝐹 “ 𝑠)))))

Distinct variable groups: 𝐴,𝑠 𝐹,𝑠 𝐺,𝑠 𝑅,𝑠 𝑆,𝑠 𝑁,𝑠 𝜑,𝑠

Proof of Theorem erdsze2lem2 Dummy variables 𝑡 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		erdsze2lem.n	. . . . 5 ⊢ 𝑁 = ((𝑅 − 1) · (𝑆 − 1))
2		erdsze2.r	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ ℕ)
3		nnm1nn0 11941	. . . . . . 7 ⊢ (𝑅 ∈ ℕ → (𝑅 − 1) ∈ ℕ₀)
4	2, 3	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑅 − 1) ∈ ℕ₀)
5		erdsze2.s	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ ℕ)
6		nnm1nn0 11941	. . . . . . 7 ⊢ (𝑆 ∈ ℕ → (𝑆 − 1) ∈ ℕ₀)
7	5, 6	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑆 − 1) ∈ ℕ₀)
8	4, 7	nn0mulcld 11963	. . . . 5 ⊢ (𝜑 → ((𝑅 − 1) · (𝑆 − 1)) ∈ ℕ₀)
9	1, 8	eqeltrid 2919	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
10		nn0p1nn 11939	. . . 4 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ ℕ)
11	9, 10	syl 17	. . 3 ⊢ (𝜑 → (𝑁 + 1) ∈ ℕ)
12		erdsze2.f	. . . 4 ⊢ (𝜑 → 𝐹:𝐴–1-1→ℝ)
13		erdsze2lem.g	. . . 4 ⊢ (𝜑 → 𝐺:(1...(𝑁 + 1))–1-1→𝐴)
14		f1co 6587	. . . 4 ⊢ ((𝐹:𝐴–1-1→ℝ ∧ 𝐺:(1...(𝑁 + 1))–1-1→𝐴) → (𝐹 ∘ 𝐺):(1...(𝑁 + 1))–1-1→ℝ)
15	12, 13, 14	syl2anc 586	. . 3 ⊢ (𝜑 → (𝐹 ∘ 𝐺):(1...(𝑁 + 1))–1-1→ℝ)
16	9	nn0red 11959	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℝ)
17	16	ltp1d 11572	. . . 4 ⊢ (𝜑 → 𝑁 < (𝑁 + 1))
18	1, 17	eqbrtrrid 5104	. . 3 ⊢ (𝜑 → ((𝑅 − 1) · (𝑆 − 1)) < (𝑁 + 1))
19	11, 15, 2, 5, 18	erdsze 32451	. 2 ⊢ (𝜑 → ∃𝑡 ∈ 𝒫 (1...(𝑁 + 1))((𝑅 ≤ (♯‘𝑡) ∧ ((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡))) ∨ (𝑆 ≤ (♯‘𝑡) ∧ ((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , ^◡ < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡)))))
20		velpw 4546	. . . 4 ⊢ (𝑡 ∈ 𝒫 (1...(𝑁 + 1)) ↔ 𝑡 ⊆ (1...(𝑁 + 1)))
21		imassrn 5942	. . . . . . . 8 ⊢ (𝐺 “ 𝑡) ⊆ ran 𝐺
22		f1f 6577	. . . . . . . . . 10 ⊢ (𝐺:(1...(𝑁 + 1))–1-1→𝐴 → 𝐺:(1...(𝑁 + 1))⟶𝐴)
23	13, 22	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐺:(1...(𝑁 + 1))⟶𝐴)
24	23	frnd 6523	. . . . . . . 8 ⊢ (𝜑 → ran 𝐺 ⊆ 𝐴)
25	21, 24	sstrid 3980	. . . . . . 7 ⊢ (𝜑 → (𝐺 “ 𝑡) ⊆ 𝐴)
26		erdsze2.a	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
27		reex 10630	. . . . . . . . 9 ⊢ ℝ ∈ V
28		ssexg 5229	. . . . . . . . 9 ⊢ ((𝐴 ⊆ ℝ ∧ ℝ ∈ V) → 𝐴 ∈ V)
29	26, 27, 28	sylancl 588	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ V)
30		elpw2g 5249	. . . . . . . 8 ⊢ (𝐴 ∈ V → ((𝐺 “ 𝑡) ∈ 𝒫 𝐴 ↔ (𝐺 “ 𝑡) ⊆ 𝐴))
31	29, 30	syl 17	. . . . . . 7 ⊢ (𝜑 → ((𝐺 “ 𝑡) ∈ 𝒫 𝐴 ↔ (𝐺 “ 𝑡) ⊆ 𝐴))
32	25, 31	mpbird 259	. . . . . 6 ⊢ (𝜑 → (𝐺 “ 𝑡) ∈ 𝒫 𝐴)
33	32	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (𝐺 “ 𝑡) ∈ 𝒫 𝐴)
34		vex 3499	. . . . . . . . . . . 12 ⊢ 𝑡 ∈ V
35	34	f1imaen 8573	. . . . . . . . . . 11 ⊢ ((𝐺:(1...(𝑁 + 1))–1-1→𝐴 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (𝐺 “ 𝑡) ≈ 𝑡)
36	13, 35	sylan 582	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (𝐺 “ 𝑡) ≈ 𝑡)
37		fzfid 13344	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (1...(𝑁 + 1)) ∈ Fin)
38		simpr 487	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → 𝑡 ⊆ (1...(𝑁 + 1)))
39		ssfi 8740	. . . . . . . . . . . . 13 ⊢ (((1...(𝑁 + 1)) ∈ Fin ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → 𝑡 ∈ Fin)
40	37, 38, 39	syl2anc 586	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → 𝑡 ∈ Fin)
41		enfii 8737	. . . . . . . . . . . 12 ⊢ ((𝑡 ∈ Fin ∧ (𝐺 “ 𝑡) ≈ 𝑡) → (𝐺 “ 𝑡) ∈ Fin)
42	40, 36, 41	syl2anc 586	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (𝐺 “ 𝑡) ∈ Fin)
43		hashen 13710	. . . . . . . . . . 11 ⊢ (((𝐺 “ 𝑡) ∈ Fin ∧ 𝑡 ∈ Fin) → ((♯‘(𝐺 “ 𝑡)) = (♯‘𝑡) ↔ (𝐺 “ 𝑡) ≈ 𝑡))
44	42, 40, 43	syl2anc 586	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → ((♯‘(𝐺 “ 𝑡)) = (♯‘𝑡) ↔ (𝐺 “ 𝑡) ≈ 𝑡))
45	36, 44	mpbird 259	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (♯‘(𝐺 “ 𝑡)) = (♯‘𝑡))
46	45	breq2d 5080	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (𝑅 ≤ (♯‘(𝐺 “ 𝑡)) ↔ 𝑅 ≤ (♯‘𝑡)))
47	46	biimprd 250	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (𝑅 ≤ (♯‘𝑡) → 𝑅 ≤ (♯‘(𝐺 “ 𝑡))))
48		erdsze2lem.i	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐺 Isom < , < ((1...(𝑁 + 1)), ran 𝐺))
49	48	ad2antrr 724	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) ∧ (𝑥 ∈ 𝑡 ∧ 𝑦 ∈ 𝑡)) → 𝐺 Isom < , < ((1...(𝑁 + 1)), ran 𝐺))
50	38	adantr 483	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) ∧ (𝑥 ∈ 𝑡 ∧ 𝑦 ∈ 𝑡)) → 𝑡 ⊆ (1...(𝑁 + 1)))
51		simprl 769	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) ∧ (𝑥 ∈ 𝑡 ∧ 𝑦 ∈ 𝑡)) → 𝑥 ∈ 𝑡)
52	50, 51	sseldd 3970	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) ∧ (𝑥 ∈ 𝑡 ∧ 𝑦 ∈ 𝑡)) → 𝑥 ∈ (1...(𝑁 + 1)))
53		simprr 771	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) ∧ (𝑥 ∈ 𝑡 ∧ 𝑦 ∈ 𝑡)) → 𝑦 ∈ 𝑡)
54	50, 53	sseldd 3970	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) ∧ (𝑥 ∈ 𝑡 ∧ 𝑦 ∈ 𝑡)) → 𝑦 ∈ (1...(𝑁 + 1)))
55		isorel 7081	. . . . . . . . . . . . . 14 ⊢ ((𝐺 Isom < , < ((1...(𝑁 + 1)), ran 𝐺) ∧ (𝑥 ∈ (1...(𝑁 + 1)) ∧ 𝑦 ∈ (1...(𝑁 + 1)))) → (𝑥 < 𝑦 ↔ (𝐺‘𝑥) < (𝐺‘𝑦)))
56	49, 52, 54, 55	syl12anc 834	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) ∧ (𝑥 ∈ 𝑡 ∧ 𝑦 ∈ 𝑡)) → (𝑥 < 𝑦 ↔ (𝐺‘𝑥) < (𝐺‘𝑦)))
57	56	biimpd 231	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) ∧ (𝑥 ∈ 𝑡 ∧ 𝑦 ∈ 𝑡)) → (𝑥 < 𝑦 → (𝐺‘𝑥) < (𝐺‘𝑦)))
58	57	ralrimivva 3193	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥 < 𝑦 → (𝐺‘𝑥) < (𝐺‘𝑦)))
59		elfznn 12939	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 ∈ (1...(𝑁 + 1)) → 𝑡 ∈ ℕ)
60	59	nnred 11655	. . . . . . . . . . . . . . 15 ⊢ (𝑡 ∈ (1...(𝑁 + 1)) → 𝑡 ∈ ℝ)
61	60	ssriv 3973	. . . . . . . . . . . . . 14 ⊢ (1...(𝑁 + 1)) ⊆ ℝ
62	61	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (1...(𝑁 + 1)) ⊆ ℝ)
63		ltso 10723	. . . . . . . . . . . . 13 ⊢ < Or ℝ
64		soss 5495	. . . . . . . . . . . . 13 ⊢ ((1...(𝑁 + 1)) ⊆ ℝ → ( < Or ℝ → < Or (1...(𝑁 + 1))))
65	62, 63, 64	mpisyl 21	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → < Or (1...(𝑁 + 1)))
66	26	adantr 483	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → 𝐴 ⊆ ℝ)
67		soss 5495	. . . . . . . . . . . . 13 ⊢ (𝐴 ⊆ ℝ → ( < Or ℝ → < Or 𝐴))
68	66, 63, 67	mpisyl 21	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → < Or 𝐴)
69	23	adantr 483	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → 𝐺:(1...(𝑁 + 1))⟶𝐴)
70		soisores 7082	. . . . . . . . . . . 12 ⊢ ((( < Or (1...(𝑁 + 1)) ∧ < Or 𝐴) ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ 𝑡 ⊆ (1...(𝑁 + 1)))) → ((𝐺 ↾ 𝑡) Isom < , < (𝑡, (𝐺 “ 𝑡)) ↔ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥 < 𝑦 → (𝐺‘𝑥) < (𝐺‘𝑦))))
71	65, 68, 69, 38, 70	syl22anc 836	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → ((𝐺 ↾ 𝑡) Isom < , < (𝑡, (𝐺 “ 𝑡)) ↔ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥 < 𝑦 → (𝐺‘𝑥) < (𝐺‘𝑦))))
72	58, 71	mpbird 259	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (𝐺 ↾ 𝑡) Isom < , < (𝑡, (𝐺 “ 𝑡)))
73		isocnv 7085	. . . . . . . . . 10 ⊢ ((𝐺 ↾ 𝑡) Isom < , < (𝑡, (𝐺 “ 𝑡)) → ^◡(𝐺 ↾ 𝑡) Isom < , < ((𝐺 “ 𝑡), 𝑡))
74	72, 73	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → ^◡(𝐺 ↾ 𝑡) Isom < , < ((𝐺 “ 𝑡), 𝑡))
75		isotr 7091	. . . . . . . . . 10 ⊢ ((^◡(𝐺 ↾ 𝑡) Isom < , < ((𝐺 “ 𝑡), 𝑡) ∧ ((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡))) → (((𝐹 ∘ 𝐺) ↾ 𝑡) ∘ ^◡(𝐺 ↾ 𝑡)) Isom < , < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡)))
76	75	ex 415	. . . . . . . . 9 ⊢ (^◡(𝐺 ↾ 𝑡) Isom < , < ((𝐺 “ 𝑡), 𝑡) → (((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡)) → (((𝐹 ∘ 𝐺) ↾ 𝑡) ∘ ^◡(𝐺 ↾ 𝑡)) Isom < , < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡))))
77	74, 76	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡)) → (((𝐹 ∘ 𝐺) ↾ 𝑡) ∘ ^◡(𝐺 ↾ 𝑡)) Isom < , < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡))))
78		resco 6105	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∘ 𝐺) ↾ 𝑡) = (𝐹 ∘ (𝐺 ↾ 𝑡))
79	78	coeq1i 5732	. . . . . . . . . . . 12 ⊢ (((𝐹 ∘ 𝐺) ↾ 𝑡) ∘ ^◡(𝐺 ↾ 𝑡)) = ((𝐹 ∘ (𝐺 ↾ 𝑡)) ∘ ^◡(𝐺 ↾ 𝑡))
80		coass 6120	. . . . . . . . . . . 12 ⊢ ((𝐹 ∘ (𝐺 ↾ 𝑡)) ∘ ^◡(𝐺 ↾ 𝑡)) = (𝐹 ∘ ((𝐺 ↾ 𝑡) ∘ ^◡(𝐺 ↾ 𝑡)))
81	79, 80	eqtri 2846	. . . . . . . . . . 11 ⊢ (((𝐹 ∘ 𝐺) ↾ 𝑡) ∘ ^◡(𝐺 ↾ 𝑡)) = (𝐹 ∘ ((𝐺 ↾ 𝑡) ∘ ^◡(𝐺 ↾ 𝑡)))
82		f1ores 6631	. . . . . . . . . . . . . . 15 ⊢ ((𝐺:(1...(𝑁 + 1))–1-1→𝐴 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (𝐺 ↾ 𝑡):𝑡–1-1-onto→(𝐺 “ 𝑡))
83	13, 82	sylan 582	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (𝐺 ↾ 𝑡):𝑡–1-1-onto→(𝐺 “ 𝑡))
84		f1ococnv2 6643	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ↾ 𝑡):𝑡–1-1-onto→(𝐺 “ 𝑡) → ((𝐺 ↾ 𝑡) ∘ ^◡(𝐺 ↾ 𝑡)) = ( I ↾ (𝐺 “ 𝑡)))
85	83, 84	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → ((𝐺 ↾ 𝑡) ∘ ^◡(𝐺 ↾ 𝑡)) = ( I ↾ (𝐺 “ 𝑡)))
86	85	coeq2d 5735	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (𝐹 ∘ ((𝐺 ↾ 𝑡) ∘ ^◡(𝐺 ↾ 𝑡))) = (𝐹 ∘ ( I ↾ (𝐺 “ 𝑡))))
87		coires1 6119	. . . . . . . . . . . 12 ⊢ (𝐹 ∘ ( I ↾ (𝐺 “ 𝑡))) = (𝐹 ↾ (𝐺 “ 𝑡))
88	86, 87	syl6eq 2874	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (𝐹 ∘ ((𝐺 ↾ 𝑡) ∘ ^◡(𝐺 ↾ 𝑡))) = (𝐹 ↾ (𝐺 “ 𝑡)))
89	81, 88	syl5eq 2870	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (((𝐹 ∘ 𝐺) ↾ 𝑡) ∘ ^◡(𝐺 ↾ 𝑡)) = (𝐹 ↾ (𝐺 “ 𝑡)))
90		isoeq1 7072	. . . . . . . . . 10 ⊢ ((((𝐹 ∘ 𝐺) ↾ 𝑡) ∘ ^◡(𝐺 ↾ 𝑡)) = (𝐹 ↾ (𝐺 “ 𝑡)) → ((((𝐹 ∘ 𝐺) ↾ 𝑡) ∘ ^◡(𝐺 ↾ 𝑡)) Isom < , < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡))))
91	89, 90	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → ((((𝐹 ∘ 𝐺) ↾ 𝑡) ∘ ^◡(𝐺 ↾ 𝑡)) Isom < , < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡))))
92		imaco 6106	. . . . . . . . . 10 ⊢ ((𝐹 ∘ 𝐺) “ 𝑡) = (𝐹 “ (𝐺 “ 𝑡))
93		isoeq5 7076	. . . . . . . . . 10 ⊢ (((𝐹 ∘ 𝐺) “ 𝑡) = (𝐹 “ (𝐺 “ 𝑡)) → ((𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡)))))
94	92, 93	ax-mp 5	. . . . . . . . 9 ⊢ ((𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡))))
95	91, 94	syl6bb 289	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → ((((𝐹 ∘ 𝐺) ↾ 𝑡) ∘ ^◡(𝐺 ↾ 𝑡)) Isom < , < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡)))))
96	77, 95	sylibd 241	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡)) → (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡)))))
97	47, 96	anim12d 610	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → ((𝑅 ≤ (♯‘𝑡) ∧ ((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡))) → (𝑅 ≤ (♯‘(𝐺 “ 𝑡)) ∧ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡))))))
98	45	breq2d 5080	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (𝑆 ≤ (♯‘(𝐺 “ 𝑡)) ↔ 𝑆 ≤ (♯‘𝑡)))
99	98	biimprd 250	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (𝑆 ≤ (♯‘𝑡) → 𝑆 ≤ (♯‘(𝐺 “ 𝑡))))
100		isotr 7091	. . . . . . . . . 10 ⊢ ((^◡(𝐺 ↾ 𝑡) Isom < , < ((𝐺 “ 𝑡), 𝑡) ∧ ((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , ^◡ < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡))) → (((𝐹 ∘ 𝐺) ↾ 𝑡) ∘ ^◡(𝐺 ↾ 𝑡)) Isom < , ^◡ < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡)))
101	100	ex 415	. . . . . . . . 9 ⊢ (^◡(𝐺 ↾ 𝑡) Isom < , < ((𝐺 “ 𝑡), 𝑡) → (((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , ^◡ < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡)) → (((𝐹 ∘ 𝐺) ↾ 𝑡) ∘ ^◡(𝐺 ↾ 𝑡)) Isom < , ^◡ < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡))))
102	74, 101	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , ^◡ < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡)) → (((𝐹 ∘ 𝐺) ↾ 𝑡) ∘ ^◡(𝐺 ↾ 𝑡)) Isom < , ^◡ < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡))))
103		isoeq1 7072	. . . . . . . . . 10 ⊢ ((((𝐹 ∘ 𝐺) ↾ 𝑡) ∘ ^◡(𝐺 ↾ 𝑡)) = (𝐹 ↾ (𝐺 “ 𝑡)) → ((((𝐹 ∘ 𝐺) ↾ 𝑡) ∘ ^◡(𝐺 ↾ 𝑡)) Isom < , ^◡ < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ^◡ < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡))))
104	89, 103	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → ((((𝐹 ∘ 𝐺) ↾ 𝑡) ∘ ^◡(𝐺 ↾ 𝑡)) Isom < , ^◡ < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ^◡ < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡))))
105		isoeq5 7076	. . . . . . . . . 10 ⊢ (((𝐹 ∘ 𝐺) “ 𝑡) = (𝐹 “ (𝐺 “ 𝑡)) → ((𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ^◡ < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ^◡ < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡)))))
106	92, 105	ax-mp 5	. . . . . . . . 9 ⊢ ((𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ^◡ < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ^◡ < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡))))
107	104, 106	syl6bb 289	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → ((((𝐹 ∘ 𝐺) ↾ 𝑡) ∘ ^◡(𝐺 ↾ 𝑡)) Isom < , ^◡ < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ^◡ < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡)))))
108	102, 107	sylibd 241	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , ^◡ < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡)) → (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ^◡ < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡)))))
109	99, 108	anim12d 610	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → ((𝑆 ≤ (♯‘𝑡) ∧ ((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , ^◡ < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡))) → (𝑆 ≤ (♯‘(𝐺 “ 𝑡)) ∧ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ^◡ < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡))))))
110	97, 109	orim12d 961	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (((𝑅 ≤ (♯‘𝑡) ∧ ((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡))) ∨ (𝑆 ≤ (♯‘𝑡) ∧ ((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , ^◡ < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡)))) → ((𝑅 ≤ (♯‘(𝐺 “ 𝑡)) ∧ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡)))) ∨ (𝑆 ≤ (♯‘(𝐺 “ 𝑡)) ∧ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ^◡ < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡)))))))
111		fveq2 6672	. . . . . . . . 9 ⊢ (𝑠 = (𝐺 “ 𝑡) → (♯‘𝑠) = (♯‘(𝐺 “ 𝑡)))
112	111	breq2d 5080	. . . . . . . 8 ⊢ (𝑠 = (𝐺 “ 𝑡) → (𝑅 ≤ (♯‘𝑠) ↔ 𝑅 ≤ (♯‘(𝐺 “ 𝑡))))
113		reseq2 5850	. . . . . . . . . 10 ⊢ (𝑠 = (𝐺 “ 𝑡) → (𝐹 ↾ 𝑠) = (𝐹 ↾ (𝐺 “ 𝑡)))
114		isoeq1 7072	. . . . . . . . . 10 ⊢ ((𝐹 ↾ 𝑠) = (𝐹 ↾ (𝐺 “ 𝑡)) → ((𝐹 ↾ 𝑠) Isom < , < (𝑠, (𝐹 “ 𝑠)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < (𝑠, (𝐹 “ 𝑠))))
115	113, 114	syl 17	. . . . . . . . 9 ⊢ (𝑠 = (𝐺 “ 𝑡) → ((𝐹 ↾ 𝑠) Isom < , < (𝑠, (𝐹 “ 𝑠)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < (𝑠, (𝐹 “ 𝑠))))
116		isoeq4 7075	. . . . . . . . 9 ⊢ (𝑠 = (𝐺 “ 𝑡) → ((𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < (𝑠, (𝐹 “ 𝑠)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < ((𝐺 “ 𝑡), (𝐹 “ 𝑠))))
117		imaeq2 5927	. . . . . . . . . 10 ⊢ (𝑠 = (𝐺 “ 𝑡) → (𝐹 “ 𝑠) = (𝐹 “ (𝐺 “ 𝑡)))
118		isoeq5 7076	. . . . . . . . . 10 ⊢ ((𝐹 “ 𝑠) = (𝐹 “ (𝐺 “ 𝑡)) → ((𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < ((𝐺 “ 𝑡), (𝐹 “ 𝑠)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡)))))
119	117, 118	syl 17	. . . . . . . . 9 ⊢ (𝑠 = (𝐺 “ 𝑡) → ((𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < ((𝐺 “ 𝑡), (𝐹 “ 𝑠)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡)))))
120	115, 116, 119	3bitrd 307	. . . . . . . 8 ⊢ (𝑠 = (𝐺 “ 𝑡) → ((𝐹 ↾ 𝑠) Isom < , < (𝑠, (𝐹 “ 𝑠)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡)))))
121	112, 120	anbi12d 632	. . . . . . 7 ⊢ (𝑠 = (𝐺 “ 𝑡) → ((𝑅 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , < (𝑠, (𝐹 “ 𝑠))) ↔ (𝑅 ≤ (♯‘(𝐺 “ 𝑡)) ∧ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡))))))
122	111	breq2d 5080	. . . . . . . 8 ⊢ (𝑠 = (𝐺 “ 𝑡) → (𝑆 ≤ (♯‘𝑠) ↔ 𝑆 ≤ (♯‘(𝐺 “ 𝑡))))
123		isoeq1 7072	. . . . . . . . . 10 ⊢ ((𝐹 ↾ 𝑠) = (𝐹 ↾ (𝐺 “ 𝑡)) → ((𝐹 ↾ 𝑠) Isom < , ^◡ < (𝑠, (𝐹 “ 𝑠)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ^◡ < (𝑠, (𝐹 “ 𝑠))))
124	113, 123	syl 17	. . . . . . . . 9 ⊢ (𝑠 = (𝐺 “ 𝑡) → ((𝐹 ↾ 𝑠) Isom < , ^◡ < (𝑠, (𝐹 “ 𝑠)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ^◡ < (𝑠, (𝐹 “ 𝑠))))
125		isoeq4 7075	. . . . . . . . 9 ⊢ (𝑠 = (𝐺 “ 𝑡) → ((𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ^◡ < (𝑠, (𝐹 “ 𝑠)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ^◡ < ((𝐺 “ 𝑡), (𝐹 “ 𝑠))))
126		isoeq5 7076	. . . . . . . . . 10 ⊢ ((𝐹 “ 𝑠) = (𝐹 “ (𝐺 “ 𝑡)) → ((𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ^◡ < ((𝐺 “ 𝑡), (𝐹 “ 𝑠)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ^◡ < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡)))))
127	117, 126	syl 17	. . . . . . . . 9 ⊢ (𝑠 = (𝐺 “ 𝑡) → ((𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ^◡ < ((𝐺 “ 𝑡), (𝐹 “ 𝑠)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ^◡ < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡)))))
128	124, 125, 127	3bitrd 307	. . . . . . . 8 ⊢ (𝑠 = (𝐺 “ 𝑡) → ((𝐹 ↾ 𝑠) Isom < , ^◡ < (𝑠, (𝐹 “ 𝑠)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ^◡ < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡)))))
129	122, 128	anbi12d 632	. . . . . . 7 ⊢ (𝑠 = (𝐺 “ 𝑡) → ((𝑆 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , ^◡ < (𝑠, (𝐹 “ 𝑠))) ↔ (𝑆 ≤ (♯‘(𝐺 “ 𝑡)) ∧ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ^◡ < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡))))))
130	121, 129	orbi12d 915	. . . . . 6 ⊢ (𝑠 = (𝐺 “ 𝑡) → (((𝑅 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , < (𝑠, (𝐹 “ 𝑠))) ∨ (𝑆 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , ^◡ < (𝑠, (𝐹 “ 𝑠)))) ↔ ((𝑅 ≤ (♯‘(𝐺 “ 𝑡)) ∧ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡)))) ∨ (𝑆 ≤ (♯‘(𝐺 “ 𝑡)) ∧ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ^◡ < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡)))))))
131	130	rspcev 3625	. . . . 5 ⊢ (((𝐺 “ 𝑡) ∈ 𝒫 𝐴 ∧ ((𝑅 ≤ (♯‘(𝐺 “ 𝑡)) ∧ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡)))) ∨ (𝑆 ≤ (♯‘(𝐺 “ 𝑡)) ∧ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ^◡ < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡)))))) → ∃𝑠 ∈ 𝒫 𝐴((𝑅 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , < (𝑠, (𝐹 “ 𝑠))) ∨ (𝑆 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , ^◡ < (𝑠, (𝐹 “ 𝑠)))))
132	33, 110, 131	syl6an 682	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (((𝑅 ≤ (♯‘𝑡) ∧ ((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡))) ∨ (𝑆 ≤ (♯‘𝑡) ∧ ((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , ^◡ < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡)))) → ∃𝑠 ∈ 𝒫 𝐴((𝑅 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , < (𝑠, (𝐹 “ 𝑠))) ∨ (𝑆 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , ^◡ < (𝑠, (𝐹 “ 𝑠))))))
133	20, 132	sylan2b 595	. . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝒫 (1...(𝑁 + 1))) → (((𝑅 ≤ (♯‘𝑡) ∧ ((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡))) ∨ (𝑆 ≤ (♯‘𝑡) ∧ ((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , ^◡ < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡)))) → ∃𝑠 ∈ 𝒫 𝐴((𝑅 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , < (𝑠, (𝐹 “ 𝑠))) ∨ (𝑆 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , ^◡ < (𝑠, (𝐹 “ 𝑠))))))
134	133	rexlimdva 3286	. 2 ⊢ (𝜑 → (∃𝑡 ∈ 𝒫 (1...(𝑁 + 1))((𝑅 ≤ (♯‘𝑡) ∧ ((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡))) ∨ (𝑆 ≤ (♯‘𝑡) ∧ ((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , ^◡ < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡)))) → ∃𝑠 ∈ 𝒫 𝐴((𝑅 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , < (𝑠, (𝐹 “ 𝑠))) ∨ (𝑆 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , ^◡ < (𝑠, (𝐹 “ 𝑠))))))
135	19, 134	mpd 15	1 ⊢ (𝜑 → ∃𝑠 ∈ 𝒫 𝐴((𝑅 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , < (𝑠, (𝐹 “ 𝑠))) ∨ (𝑆 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , ^◡ < (𝑠, (𝐹 “ 𝑠)))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 = wceq 1537 ∈ wcel 2114 ∀wral 3140 ∃wrex 3141 Vcvv 3496 ⊆ wss 3938 𝒫 cpw 4541 class class class wbr 5068 I cid 5461 Or wor 5475 ^◡ccnv 5556 ran crn 5558 ↾ cres 5559 “ cima 5560 ∘ ccom 5561 ⟶wf 6353 –1-1→wf1 6354 –1-1-onto→wf1o 6356 ‘cfv 6357 Isom wiso 6358 (class class class)co 7158 ≈ cen 8508 Fincfn 8511 ℝcr 10538 1c1 10540 + caddc 10542 · cmul 10544 < clt 10677 ≤ cle 10678 − cmin 10872 ℕcn 11640 ℕ₀cn0 11900 ...cfz 12895 ♯chash 13693

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-sup 8908 df-dju 9332 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-n0 11901 df-xnn0 11971 df-z 11985 df-uz 12247 df-fz 12896 df-hash 13694

This theorem is referenced by: erdsze2 32454

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